(* Inversion lemmas with context-sensitive stringly rt-normalizing terms ****)
-lemma fsb_inv_csx: ∀h,G,L,T. ≥[h] 𝐒⦃G,L,T⦄ → ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄.
+lemma fsb_inv_csx (h):
+ ∀G,L,T. ≥𝐒[h] ❪G,L,T❫ → ❪G,L❫ ⊢ ⬈*𝐒[h] T.
#h #G #L #T #H @(fsb_ind_alt … H) -G -L -T /5 width=1 by csx_intro, fpb_cpx/
qed-.
(* Propreties with context-sensitive stringly rt-normalizing terms **********)
-lemma csx_fsb_fpbs: ∀h,G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1,L1,T1⦄ ≥[h] ⦃G2,L2,T2⦄ → ≥[h] 𝐒⦃G2,L2,T2⦄.
+lemma csx_fsb_fpbs (h):
+ ∀G1,L1,T1. ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
+ ∀G2,L2,T2. ❪G1,L1,T1❫ ≥[h] ❪G2,L2,T2❫ → ≥𝐒[h] ❪G2,L2,T2❫.
#h #G1 #L1 #T1 #H @(csx_ind … H) -T1
#T1 #HT1 #IHc #G2 #L2 #T2 @(fqup_wf_ind (Ⓣ) … G2 L2 T2) -G2 -L2 -T2
-#G0 #L0 #T0 #IHu #H10
+#G0 #L0 #T0 #IHu #H10
lapply (fpbs_csx_conf … H10) // -HT1 #HT0
generalize in match IHu; -IHu generalize in match H10; -H10
@(rsx_ind … (csx_rsx … HT0)) -L0
#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
| #T2 #HT02 #HnT02
- elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0
+ elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0
/3 width=4 by/
| #L2 #HL02 #HnL02 @(IHd … HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
[ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
| #G3 #L3 #T3 #H03 #_
elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
- elim (tdeq_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
- [ elim (tdeq_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
- /4 width=7 by fsb_fpbs_trans, tdeq_rdeq_lpx_fpbs, fpbs_fqup_trans/
- | elim (cpxs_tdneq_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
- elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
- /3 width=14 by fpbs_cpxs_tdeq_fqup_lpx_trans/
+ elim (teqx_dec T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
+ [ elim (teqx_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
+ /4 width=7 by fsb_fpbs_trans, teqx_reqx_lpx_fpbs, fpbs_fqup_trans/
+ | elim (cpxs_tneqx_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
+ elim (fpbs_cpx_tneqx_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
+ /3 width=14 by fpbs_cpxs_teqx_fqup_lpx_trans/
]
]
]
qed.
-lemma csx_fsb: ∀h,G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → ≥[h] 𝐒⦃G,L,T⦄.
+lemma csx_fsb (h):
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → ≥𝐒[h] ❪G,L,T❫.
/2 width=5 by csx_fsb_fpbs/ qed.
(* Advanced eliminators *****************************************************)
-lemma csx_ind_fpb: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀G2,L2,T2. ⦃G1,L1,T1⦄ ≻[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
+lemma csx_ind_fpb (h) (Q:relation3 …):
+ (∀G1,L1,T1.
+ ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ ≻[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_alt/ qed-.
-lemma csx_ind_fpbg: ∀h. ∀Q:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1,L1⦄ ⊢ ⬈*[h] 𝐒⦃T1⦄ →
- (∀G2,L2,T2. ⦃G1,L1,T1⦄ >[h] ⦃G2,L2,T2⦄ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) →
- ∀G,L,T. ⦃G,L⦄ ⊢ ⬈*[h] 𝐒⦃T⦄ → Q G L T.
+lemma csx_ind_fpbg (h) (Q:relation3 …):
+ (∀G1,L1,T1.
+ ❪G1,L1❫ ⊢ ⬈*𝐒[h] T1 →
+ (∀G2,L2,T2. ❪G1,L1,T1❫ >[h] ❪G2,L2,T2❫ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) →
+ ∀G,L,T. ❪G,L❫ ⊢ ⬈*𝐒[h] T → Q G L T.
/4 width=4 by fsb_inv_csx, csx_fsb, fsb_ind_fpbg/ qed-.